A008616 -id:A008616 - cp's OEIS frontend

A000115 Denumerants: Expansion of 1/((1-x)(1-x^2)(1-x^5)).

1, 1, 2, 2, 3, 4, 5, 6, 7, 8, 10, 11, 13, 14, 16, 18, 20, 22, 24, 26, 29, 31, 34, 36, 39, 42, 45, 48, 51, 54, 58, 61, 65, 68, 72, 76, 80, 84, 88, 92, 97, 101, 106, 110, 115, 120, 125, 130, 135, 140, 146, 151, 157, 162, 168, 174, 180, 186, 192, 198, 205, 211, 218, 224, 231, 238

Offset: 0

N. J. A. Sloane

Number of partitions of n into parts 1, 2, or 5.

First differences are in A008616. First differences of A001304. Pairwise sums of A008720.

			G.f. = 1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 5*x^6 + 6*x^7 + 7*x^8 + ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 120, D(n;1,2,5).
M. Jeger, Ein partitions problem ..., Elemente de Math., 13 (1958), 97-120.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 152.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Mohammed L. Nadji, Moussa Ahmia, Daniel F. Checa, and José L. Ramírez, Arndt Compositions with Restricted Parts, Palindromes, and Colored Variants, J. Int. Seq. (2025) Vol. 28, Issue 3, Article 25.3.6. See p. 15.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1,0,1,-1,-1,1)

Cf. A001304, A008616, A008720.

[Round((n+4)^2/20): n in [0..70]]; // Vincenzo Librandi, Jun 23 2011

1/((1-x)*(1-x^2)*(1-x^5)): seq(coeff(series(%, x, n+1), x, n), n=0..65);
# next Maple program:
s:=proc(n) if n mod 5 = 0 then RETURN(1); fi; if n mod 5 = 1 then RETURN(0); fi; if n mod 5 = 2 then RETURN(1); fi; if n mod 5 = 3 then RETURN(-1); fi; if n mod 5 = 4 then RETURN(-1); fi; end: f:=n->(2*n^2+16*n+27+5*(-1)^n+8*s(n))/40: seq(f(n), n=0..65);  # from Jeger's paper

nn=50;CoefficientList[Series[1/(1-x)/(1-x^2)/(1-x^5),{x,0,nn}],x]  (* Geoffrey Critzer, Jan 20 2013 *)
LinearRecurrence[{1,1,-1,0,1,-1,-1,1},{1,1,2,2,3,4,5,6},70] (* Harvey P. Dale, Sep 27 2019 *)

a(n)=(n^2+8*n+26)\20 \\ Charles R Greathouse IV, Jun 23 2011

a(n) = round((n+4)^2/20).

a(n) = a(-8 - n) for all n in Z. - Michael Somos, May 28 2014

A025795 Expansion of 1/((1-x^2)(1-x^3)(1-x^5)).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 5, 5, 6, 7, 7, 8, 9, 9, 11, 11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 21, 23, 24, 25, 27, 28, 29, 31, 32, 34, 35, 37, 38, 40, 42, 43, 45, 47, 48, 51, 52, 54, 56, 58, 60, 62, 64, 66, 68, 71, 72, 75, 77, 79, 82, 84, 86, 89, 91, 94, 96, 99, 101, 104

Offset: 0

N. J. A. Sloane

a(n) is the number of ways to pay n dollars with coins of two, three and five dollars. E.g., a(0)=1 because there is one way to pay: with no coin; a(1)=0 no possibility; a(2)=1 (2=1*2); a(3)=1 (3=1*3); a(4)=1 (4=2*2) a(5)=2 (5=3+2=1*5) ... - Richard Choulet, Jan 20 2008

a(n) is the number of partitions of n into parts 2, 3, and 5. See the preceding comment by R. Choulet. - Wolfdieter Lang, Mar 15 2012

			G.f. = 1 + x^2 + x^3 + x^4 + 2*x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 3*x^9 + 4*x^10 + ...

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Milan Janjic, On Linear Recurrence Equations Arising from Compositions of Positive Integers, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.7.
Index entries for linear recurrences with constant coefficients, signature (0,1,1,0,0,0,-1,-1,0,1).

Cf. A008676, A078495, A103221.

a[ n_] := Quotient[n^2 + 10 n + 1 - 13 Mod[n, 2], 60] + 1; (* Michael Somos, Nov 17 2017 *)

{a(n) = (n^2 + 10*n + 1 - n%2 * 13) \60 + 1} /* Michael Somos, Feb 05 2008 */

G.f.: 1/((1-x^2)*(1-x^3)*(1-x^5)).
Let [b(1); b(2); ...; b(p)] denote a periodic sequence: e.g., [0; 1] defines the sequence c such that c(0)=c(2)=...=c(2*k)=0 and c(1)=c(3)=...=c(2*k+1)=1. Then a(n)=0.25*[0; 1] - (1/3)*[1; 0; 0] + (1/5)*[0; 1; 1; 0; 3] + ((n+1)*(n+2)/60) + (7*(n+1)/60). - Richard Choulet, Jan 20 2008
If ||A|| is the nearest number to A (A not a half-integer) we also have a(n) = ||((n+1)*(n+9)/60) + (1/5)[0; 1; 1; 0; 3]. - Richard Choulet, Jan 20 2008
a(n) = 77/360 + 7*(n+1)/60 + (n+2)*(n+1)/60 + (-1)^n/8 - (2/9)*cos(2*(n+2)*Pi/3) + (4/(5*sqrt(5)+25))*cos(2*n*Pi/5) - (4/(5*sqrt(5)-25))*cos(4*n*Pi/5). - Richard Choulet, Jan 20 2008
Euler transform of length 5 sequence [0, 1, 1, 0, 1]. - Michael Somos, Feb 05 2008
a(n) = a(-10-n) for all n in Z. - Michael Somos, Feb 25 2008
a(n) - a(n-2) = A008676(n). a(n) - a(n-5) = A103221(n) = A008615(n+2). A078495(n) = 2^(a(n-7) + a(n-9)) * 3^a(n-8) for all n in Z. - Michael Somos, Nov 17 2017, corrected Jun 23 2021
a(n)-a(n-3) = A008616(n). - R. J. Mathar, Jun 23 2021
a(n) = floor((n^2 + 10*n + 6*(9+(-1)^n))/60). - Hoang Xuan Thanh, Jun 15 2025

A165190 G.f.: 1/((1-x^4)*(1-x^5)).

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 2, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 4, 4, 3, 3, 4, 4, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 4, 5, 5, 4, 4, 5, 5, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 5, 5, 6

Offset: 0

Alford Arnold, Sep 24 2009

A121262 convolved with A079998. The two sequences have very simple generating functions and can be mapped to the numeric partitions 4=4 and 5=5 respectively.

Number of partitions of n into parts 4 and 5. - Joerg Arndt, Aug 28 2015

Michael De Vlieger, Table of n, a(n) for n = 0..10000
A. V. Kitaev and A. Vartanian, Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter, arXiv:2304.05671 [math.CA], 2023. See p. 20.
Index to Molien series
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1,1,0,0,0,-1).

Cf. A057077, A079998, A117444, A121262.

[Floor((n+4)/4) - Floor((n+4)/5) : n in [0..100]]; // Wesley Ivan Hurt, Aug 27 2015

A165190:=n->floor((n+4)/4) - floor((n+4)/5): seq(A165190(n), n=0..100); # Wesley Ivan Hurt, Aug 27 2015

CoefficientList[Series[1/((1-x^4)(1-x^5)),{x,0,110}],x] (* or *) LinearRecurrence[{0,0,0,1,1,0,0,0,-1},{1,0,0,0,1,1,0,0,1},110] (* Harvey P. Dale, Aug 16 2012 *)
Table[Floor[(n + 4)/4] - Floor[(n + 4)/5], {n, 0, 100}] (* Wesley Ivan Hurt, Aug 27 2015 *)

1 followed by the Euler transform of the finite sequence [0,0,0,1,1].
G.f.: 1/((1-x)^2*(1+x)*(1+x^2)*(1+x+x^2+x^3+x^4)). [R. J. Mathar, Oct 07 2009]
a(n) = A117444(n+2)/5 + n/20 + 9/40 + (-1)^n/8 + A057077(n)/4. [R. J. Mathar, Oct 07 2009]
a(0)=1, a(1)=0, a(2)=0, a(3)=0, a(4)=1, a(5)=1, a(6)=0, a(7)=0, a(8)=1, a(n) = a(n-4)+a(n-5)-a(n-9), n>8. - Harvey P. Dale, Aug 16 2012
a(n) = floor((n+4)/4) - floor((n+4)/5). - Wesley Ivan Hurt, Aug 27 2015
a(n)+a(n-2) = A008616(n). - R. J. Mathar, Jun 23 2021

Removed duplicate of comment in A165188; Euler transform formula corrected - R. J. Mathar, Oct 07 2009

A246720 Number A(n,k) of partitions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 1, 0, 1, 1, 3, 0, 1, 0, 1, 1, 0, 2, 0, 3, 1, 1, 0, 1, 0, 1, 0, 2, 0, 4, 0, 1, 0, 1, 0, 1, 1, 1, 2, 1, 4, 1, 1, 0, 1, 1, 0, 1, 2, 0, 3, 0, 5, 0, 1, 0, 1, 1, 2, 0, 1, 2, 0, 3, 0, 5, 1, 1, 0

Offset: 0

Alois P. Heinz, Sep 02 2014

The first lists of distinct parts in the order given by A246688 are: 0:[], 1:[1], 2:[2], 3:[1,2], 4:[3], 5:[1,3], 6:[4], 7:[1,4], 8:[2,3], 9:[5], 10:[1,2,3], 11:[1,5], 12:[2,4], 13:[6], 14:[1,2,4], 15:[1,6], 16:[2,5], 17:[3,4], 18:[7], 19:[1,2,5], 20:[1,3,4], ... .

			Square array A(n,k) begins:
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1, 1, 1, 1,  1, ...
  0, 1, 0, 1, 0, 1, 0, 1, 0, 0,  1, 1, 0, 0,  1, ...
  0, 1, 1, 2, 0, 1, 0, 1, 1, 0,  2, 1, 1, 0,  2, ...
  0, 1, 0, 2, 1, 2, 0, 1, 1, 0,  3, 1, 0, 0,  2, ...
  0, 1, 1, 3, 0, 2, 1, 2, 1, 0,  4, 1, 2, 0,  4, ...
  0, 1, 0, 3, 0, 2, 0, 2, 1, 1,  5, 2, 0, 0,  4, ...
  0, 1, 1, 4, 1, 3, 0, 2, 2, 0,  7, 2, 2, 1,  6, ...
  0, 1, 0, 4, 0, 3, 0, 2, 1, 0,  8, 2, 0, 0,  6, ...
  0, 1, 1, 5, 0, 3, 1, 3, 2, 0, 10, 2, 3, 0,  9, ...
  0, 1, 0, 5, 1, 4, 0, 3, 2, 0, 12, 2, 0, 0,  9, ...
  0, 1, 1, 6, 0, 4, 0, 3, 2, 1, 14, 3, 3, 0, 12, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-11, 13-20, 23 give: A000007, A000012, A059841, A008619, A079978, A008620, A121262, A008621, A103221, A079998, A001399, A002266(n+5), A079979, A008642, A097992, A008616, A008679, A082784, A000115, A025767, A008676.
Main diagonal gives A246721.
Cf. A246688, A246690 (the same for compositions).

b:= proc(n, i) b(n, i):= `if`(n=0, [[]], `if`(i>n, [],
      [map(x->[i, x[]], b(n-i, i+1))[], b(n, i+1)[]]))
    end:
f:= proc() local i, l; i, l:=0, [];
      proc(n) while n>=nops(l)
        do l:=[l[], b(i, 1)[]]; i:=i+1 od; l[n+1]
      end
    end():
g:= proc(n, l) option remember; `if`(n=0, 1, `if`(l=[], 0,
      add(g(n-l[-1]*j, subsop(-1=NULL, l)), j=0..n/l[-1])))
    end:
A:= (n, k)-> g(n, f(k)):
seq(seq(A(n, d-n), n=0..d), d=0..16);

b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i > n, {}, Join[Prepend[#, i]& /@ b[n - i, i + 1], b[n, i + 1]]]];
f = Module[{i = 0, l = {}}, Function[n, While[ n >= Length[l], l = Join[l, b[i, 1]]; i++ ]; l[[n + 1]]]];
g[n_, l_] := g[n, l] = If[n == 0, 1, If[l == {}, 0, Sum[g[n - l[[-1]] j, ReplacePart[l, -1 -> Nothing]], {j, 0, n/l[[-1]]}]]];
A[n_, k_] := g[n, f[k]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 16}] // Flatten (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

A025810 Expansion of 1/((1-x^2)(1-x^5)(1-x^10)).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 3, 3, 3, 3, 3, 6, 3, 6, 3, 6, 6, 6, 6, 6, 6, 10, 6, 10, 6, 10, 10, 10, 10, 10, 10, 15, 10, 15, 10, 15, 15, 15, 15, 15, 15, 21, 15, 21, 15, 21, 21, 21, 21, 21, 21, 28, 21, 28, 21, 28, 28, 28, 28, 28, 28, 36, 28, 36, 28, 36, 36, 36, 36, 36, 36

Offset: 0

N. J. A. Sloane

Number of partitions of n into parts of size 2, 5, and 10.

a(n) is always a triangular number.

			G.f. = 1 + x^2 + x^4 + x^5 + x^6 + x^7 + x^8 + x^9 + 3*x^10 + x^11 + 3*x^12 + ...

Index entries for linear recurrences with constant coefficients, signature (0,1,0,0,1,0,-1,0,0,1,0,-1,0,0,-1,0,1).

Cf. A000008, A000217, A008616.

CoefficientList[Series[1/((1-x^2)(1-x^5)(1-x^10)), {x,0,85}], x]  (* Harvey P. Dale, Apr 06 2011 *)
a[ n_] := Module[ {m = Mod[n, 10], k}, k = n - m; If[ m == 1 || m == 3, k -= 10]; k (k + 30) / 200 + 1]; (* Michael Somos, Aug 16 2016 *)

{a(n) = if( n<-16, a(-17 - n), polcoeff( 1 / ((1 - x^2) * (1 - x^5) * (1 - x^10)) + x * O(x^n), n))}; \\ Michael Somos, Mar 18 2012

{a(n) = my(m = n%10); n -= m; if( m==1 || m==3, n -= 10); n * (n + 30) / 200 + 1}; \\ Michael Somos, Aug 16 2016

a(n) = (n^2 + 17*n + (5*n+22)*(-1)^n + 200 + 4*n*[2,-1,1,-2,0][n%5+1])\200 \\ Hoang Xuan Thanh, Aug 28 2025

G.f.: 1/((1-x^2)(1-x^5)(1-x^10)).
Euler transform of length 10 sequence [ 0, 1, 0, 0, 1, 0, 0, 0, 0, 1]. - Michael Somos, Mar 18 2012
a(n) = a(-17 - n) = a(n - 10) + A008616(n) for all n in Z. - Michael Somos, Mar 18 2012
a(n) = A000217( A008616(n) ) = A000008(n) - A000008(n - 1). - Michael Somos, Dec 15 2002

A025802 Expansion of 1/((1-x^2)(1-x^4)(1-x^5)).

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 2, 1, 3, 2, 4, 2, 5, 3, 6, 4, 7, 5, 8, 6, 10, 7, 11, 8, 13, 10, 14, 11, 16, 13, 18, 14, 20, 16, 22, 18, 24, 20, 26, 22, 29, 24, 31, 26, 34, 29, 36, 31, 39, 34, 42, 36, 45, 39, 48, 42, 51, 45, 54, 48, 58, 51

Offset: 0

N. J. A. Sloane

a(n) is the number of partitions of n into parts 2, 4, and 5. - Hoang Xuan Thanh, Jun 18 2025

Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,1,-1,-1,0,-1,0,1).

Cf. A000115.

CoefficientList[Series[1/((1-x^2)(1-x^4)(1-x^5)),{x,0,70}],x] (* Harvey P. Dale, Sep 15 2011 *)

From R. J. Mathar, Jun 23 2021: (Start)
a(n)-a(n-2) = A165190(n).
a(n)-a(n-4) = A008616(n). (End)
a(n) = floor((n^2 + n*(11+5*(-1)^n) + 53 + 27*(-1)^n)/80). - Hoang Xuan Thanh, Jun 18 2025

A177849 The number of ways of minimal weight to make change for n cents using fairly valued United States coins (copper 1-cent coin, a nickel 5-cent coin, and silver 10-cent and 25-cent coins) assuming that silver is more valuable than nickel and that nickel is more valuable than copper.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3

Offset: 0

Lee A. Newberg, May 14 2010

			For n = 51 cents, the least weight is achieved with 50 cents in silver and 1 cent in copper. The 50 cents in silver can be achieved as two 25-cent coins or five 10-cent coins; thus there are a(51) = 2 ways to make 51 cents with minimal weight.

Except for the values dependent upon nickel (i.e., a(5) through a(9) and a(15) through a(19)) this sequence can be constructed by repeating five times each term from sequence A008616.

G.f.: [1/(1-x^10)/(1-x^25)+x^5+x^15][1+x+x^2+x^3+x^4]

A206799 Based on an erroneous version of A008614.

Original entry on oeis.org

4, 1, 0, 2, 4, 3, 4, 4, 4, 5, 4, 6, 8, 7, 8, 8, 8, 9, 12, 10, 12, 15, 12, 12, 16, 17, 16, 18, 20, 19, 20, 20, 24, 25, 24, 26, 28, 27, 28, 32, 32, 33, 36, 34, 36, 39, 40, 40, 44, 45, 44

Offset: 0

N. J. A. Sloane, Feb 21 2012

dead

This is based on the formula in Burnside, Section 267, at the foot of page 363. Unfortunately there is a typo in the formula - the term with numerator 21 should have denominator (1+x)(1-x^3). This produces a sequence with 4's in the denominators. Multiplying by 4 gives a sequence of integers, shown here. This is included in the OEIS in accordance with our policy of publishing incorrect sequences together with pointers to the correct versions. - N. J. A. Sloane, Feb 21 2012

W. Burnside, Theory of Groups of Finite Order, Dover, NY, 1955, section 267, page 363

Cf. A008616.

(* expansion*)
w = Exp[I*2*Pi/7];
p[x_] = FullSimplify[ExpandAll[(4/168)*(1/(1 - x)^3 + 21/((1 - x)*(1 - x^2)) + 42/((1 - x)*(1 + x^2)) + 56/(1 - x^3) + 24/((1 - w*x)*(1 - w^2*x)*(1 - w^4*x)) + 24/((1 - w^6*x)*(1 - x*w^5)*(1 - x*w^3)))]];
a = Table[SeriesCoefficient[Series[FullSimplify[ExpandAll[p[x]]], {x, 0, 50}], n], {n, 0, 50}]
(* recursion*)
b[1] = 4; b[2] = 1; b[3] = 0; b[4] = 2; b[5] = 4; b[6] = 3;
b[7] = 4; b[8] = 4; b[9] = 4; b[10] = 5; b[11] = 4;
b[n_Integer?Positive] :=
b[n] = -489 + 11 n + n^2 - b[-11 + n] - 3 b[-10 + n] - 6 b[-9 + n] -
   9 b[-8 + n] - 11 b[-7 + n] - 12 b[-6 + n] - 12 b[-5 + n] -
   11 b[-4 + n] - 9 b[-3 + n] - 6 b[-2 + n] - 3 b[-1 + n];
Table[b[n], {n, 1, Length[a]}]

Vec((-4-x+2*x^3+x^4-2*x^5-2*x^6+2*x^7+3*x^8+2*x^9-3*x^11)/(-1+x^3*(1+x-x^7-x^8+x^11))+O(x^9)) \\ Charles R Greathouse IV, Feb 13 2012

A precise definition is: Take the generating function as given by Burnside, expand as a Taylor series, and multiply by 4.

Expansion of (-4 - x + 2 x^3 + x^4 - 2 x^5 - 2 x^6 + 2 x^7 + 3 x^8 + 2 x^9 - 3 x^11)/(-1 + x^3 (1 + x - x^7 - x^8 + x^11))

cp's OEIS Frontend

A000115 Denumerants: Expansion of 1/((1-x)*(1-x^2)*(1-x^5)).

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A025795 Expansion of 1/((1-x^2)*(1-x^3)*(1-x^5)).

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A165190 G.f.: 1/((1-x^4)*(1-x^5)).

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A246720 Number A(n,k) of partitions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A025810 Expansion of 1/((1-x^2)*(1-x^5)*(1-x^10)).

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A025802 Expansion of 1/((1-x^2)*(1-x^4)*(1-x^5)).

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A177849 The number of ways of minimal weight to make change for n cents using fairly valued United States coins (copper 1-cent coin, a nickel 5-cent coin, and silver 10-cent and 25-cent coins) assuming that silver is more valuable than nickel and that nickel is more valuable than copper.

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A000115 Denumerants: Expansion of 1/((1-x)(1-x^2)(1-x^5)).

A025795 Expansion of 1/((1-x^2)(1-x^3)(1-x^5)).

A025810 Expansion of 1/((1-x^2)(1-x^5)(1-x^10)).

A025802 Expansion of 1/((1-x^2)(1-x^4)(1-x^5)).